The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$

Question

In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using the [Riesz rearrangement inequality][2]. My question at first was whether $$G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k \, dy,$$ $k>0$, there is $x_0\in \mathbb{R}^d$ at which the integral reaches its maximum. This time it is not possible to use Riesz rearrangement inequality, as $|y|^k$, $k>0$ is non-decreasing.

In a comment of Christian Remling below, it was noted that $$(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k \, dy \rightarrow \infty \text{ as } |x|\rightarrow \infty.$$ So a question arose, whether there is any upper bound for $$\int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k \, dy,$$ $k>0$, of polynomial type, so that I can control the growth of $G(x)$, possibly multiplying by $|x|^\beta$, for some $\beta <0$. Is there any result in this direction, or any suggestions?

Answer 1

Note that, if $k\ge1$, then
$$G(x)=E|x+\sqrt{2t}Z|^k\le(|x|+\sqrt{2t}\|Z\|_k)^k,$$ by Minkowski's inequality, where $Z$ is a standard normal random vector in $\mathbb R^d$ and $\|Z\|_k:=(E|Z|^k)^{1/k}$. If $k\in(0,1)$, then
$$G(x)=E|x+\sqrt{2t}Z|^k\le|x|^k+(2t)^{k/2}\|Z\|_k^k.$$

Also, by Jensen's inequality, $$G(x)\ge |x|^k$$ if $k\ge1$, and $G(x)\ge |x|^k-(2t)^{k/2}\|Z\|_k^k$ if $k\in(0,1)$.

So, $$G(x)\sim|x|^k$$ as $|x|\to\infty$.